On a Roll Again: Analysis of a Dice Removal Game

TL;DR

This paper analyzes a dice removal game, deriving recursive and explicit formulas for the expected number of turns and variance, and models the game as the maximum of i.i.d. geometric variables, contributing new insights to a previously unstudied problem.

Contribution

It provides the first analysis of this dice removal game, including recursive and explicit formulas for expectation and variance, and connects it to geometric distributions.

Findings

Derived recursive and explicit formulas for expected turns and variance.

Established bounds for expectation and variance.

Modeled the game as the maximum of i.i.d. geometric variables.

Abstract

Suppose we have $n$ dice, each with $s$ faces (assume $s\geq n$). On the first turn, roll all of them, and remove from play those that rolled an $n$. Roll all of the remaining dice. In general, if at a certain turn you are left with $k$ dice, roll all of them and remove from play those that rolled a $k$. The game ends when you are left with no dice to roll. For $n,s \in \mathbb{N} \setminus \{0\}$ such that $s \geq n$, let $Y_n^s$ be the random variable for the number of turns to finish the game rolling $n$ dice with $s$ faces. We find recursive and non-recursive solutions for $\mathbb{E}(Y_n^{s})$ and $\mathrm{Var}(Y_n^{s})$, and bounds for both values. Moreover, we show that $Y_n^{s}$ can also be modeled as the maximum of a sequence of i.i.d. geometrically distributed random variables. Although, as far as we know, this game hasn't been studied before, similar problems have.